Given an array A of N numbers where A
i
represent the value of the (i+1)
th
node. Also given are M pair of edges where u and v represent the nodes that are connected by an edge. The task is to find the sum of the minimum element in all the connected components of the given undirected graph. If a node has no connectivity to any other node, count it as a component with one node.
Examples:
Input: a[] = {1, 6, 2, 7, 3, 8, 4, 9, 5, 10} m = 5 1 2 3 4 5 6 7 8 9 10 Output: 15 Connected components are: 1–2, 3–4, 5–6, 7–8 and 9–10 Sum of Minimum of all them : 1 + 2 + 3 + 4 + 5 = 15 Input: a[] = {2, 5, 3, 4, 8} m = 2 1 4 4 5 Output: 10
Approach:
Finding connected components for an undirected graph is an easier task. Doing either a
or
starting from every unvisited vertex will give us our connected components. Create a
visited[]
array which has initially all nodes marked as False. Iterate all the nodes, if the node is not visited, call
DFS()
function so that all the nodes connected directly or indirectly to the node are marked as visited. While visiting all the directly or indirectly connected nodes, store the minimum value of all nodes. Create a variable
sum
which stores the summation of the minimum of all these connected components. Once all the nodes are visited,
sum
will have the answer to the problem. Below is the implementation of the above approach:
// C++ program to find the sum // of the minimum elements in all // connected components of an undirected graph #include <bits/stdc++.h> using namespace std;
const int N = 10000;
vector< int > graph[N];
// Initially all nodes // marked as unvisited bool visited[N];
// DFS function that visits all // connected nodes from a given node void dfs( int node, int a[], int mini)
{ // Stores the minimum
mini = min(mini, a[node]);
// Marks node as visited
visited[node] = true ;
// Traversed in all the connected nodes
for ( int i : graph[node]) {
if (!visited[i])
dfs(i, a, mini);
}
} // Function to add the edges void addedge( int u, int v)
{ graph[u - 1].push_back(v - 1);
graph[v - 1].push_back(u - 1);
} // Function that returns the sum of all minimums // of connected componenets of graph int minimumSumConnectedComponents( int a[], int n)
{ // Initially sum is 0
int sum = 0;
// Traverse for all nodes
for ( int i = 0; i < n; i++) {
if (!visited[i]) {
int mini = a[i];
dfs(i, a, mini);
sum += mini;
}
}
// Returns the answer
return sum;
} // Driver Code int main()
{ int a[] = {1, 6, 2, 7, 3, 8, 4, 9, 5, 10};
// Add edges
addedge(1, 2);
addedge(3, 4);
addedge(5, 6);
addedge(7, 8);
addedge(9, 10);
int n = sizeof (a) / sizeof (a[0]);
// Calling Function
cout << minimumSumConnectedComponents(a, n);
return 0;
} |
import java.util.ArrayList;
import java.util.List;
public class ConnectedComponents {
static final int N = 10000 ;
static List<Integer>[] graph = new ArrayList[N];
static boolean [] visited = new boolean [N];
// DFS function that visits all connected nodes from a
// given node
static void dfs( int node, int [] a, int [] mini)
{
// Stores the minimum
mini[ 0 ] = Math.min(mini[ 0 ], a[node]);
// Marks node as visited
visited[node] = true ;
// Traversed in all the connected nodes
for ( int i : graph[node]) {
if (!visited[i]) {
dfs(i, a, mini);
}
}
}
// Function to add the edges
static void addEdge( int u, int v)
{
graph[u - 1 ].add(v - 1 );
graph[v - 1 ].add(u - 1 );
}
// Function that returns the sum of all minimums
// of connected components of graph
static int minimumSumConnectedComponents( int [] a, int n)
{
// Initially sum is 0
int sum = 0 ;
// Traverse for all nodes
for ( int i = 0 ; i < n; i++) {
if (!visited[i]) {
int [] mini = { a[i] };
dfs(i, a, mini);
sum += mini[ 0 ];
}
}
// Returns the answer
return sum;
}
// Driver Code
public static void main(String[] args)
{
for ( int i = 0 ; i < N; i++) {
graph[i] = new ArrayList<>();
}
int [] a = { 1 , 6 , 2 , 7 , 3 , 8 , 4 , 9 , 5 , 10 };
// Add edges
addEdge( 1 , 2 );
addEdge( 3 , 4 );
addEdge( 5 , 6 );
addEdge( 7 , 8 );
addEdge( 9 , 10 );
int n = a.length;
// Calling Function
System.out.println(
minimumSumConnectedComponents(a, n));
}
} |
from collections import defaultdict
N = 10000
graph = defaultdict( list )
visited = [ False ] * N
def dfs(node, a, mini):
# Stores the minimum
mini[ 0 ] = min (mini[ 0 ], a[node])
# Marks node as visited
visited[node] = True
# Traverse all connected nodes
for i in graph[node]:
if not visited[i]:
dfs(i, a, mini)
def add_edge(u, v):
graph[u - 1 ].append(v - 1 )
graph[v - 1 ].append(u - 1 )
def minimum_sum_connected_components(a, n):
# Initially sum is 0
total_sum = 0
# Traverse all nodes
for i in range (n):
if not visited[i]:
mini = [a[i]]
dfs(i, a, mini)
total_sum + = mini[ 0 ]
# Returns the answer
return total_sum
# Driver Code if __name__ = = "__main__" :
a = [ 1 , 6 , 2 , 7 , 3 , 8 , 4 , 9 , 5 , 10 ]
# Add edges
add_edge( 1 , 2 )
add_edge( 3 , 4 )
add_edge( 5 , 6 )
add_edge( 7 , 8 )
add_edge( 9 , 10 )
n = len (a)
# Calling Function
print (minimum_sum_connected_components(a, n))
|
using System;
using System.Collections.Generic;
class Program {
const int N = 10000;
static List< int >[] graph = new List< int >[ N ];
static bool [] visited = new bool [N];
// DFS function that visits all
// connected nodes from a given node
static void DFS( int node, int [] a, ref int mini)
{
// Stores the minimum
mini = Math.Min(mini, a[node]);
// Marks node as visited
visited[node] = true ;
// Traversed in all the connected nodes
foreach ( int i in graph[node])
{
if (!visited[i])
DFS(i, a, ref mini);
}
}
// Function to add the edges
static void AddEdge( int u, int v)
{
graph[u - 1].Add(v - 1);
graph[v - 1].Add(u - 1);
}
// Function that returns the sum of all minimums
// of connected components of the graph
static int MinimumSumConnectedComponents( int [] a, int n)
{
// Initially sum is 0
int sum = 0;
// Traverse for all nodes
for ( int i = 0; i < n; i++) {
if (!visited[i]) {
int mini = a[i];
DFS(i, a, ref mini);
sum += mini;
}
}
// Returns the answer
return sum;
}
// Driver Code
static void Main()
{
int [] a = { 1, 6, 2, 7, 3, 8, 4, 9, 5, 10 };
// Initialize graph
for ( int i = 0; i < N; i++) {
graph[i] = new List< int >();
}
// Add edges
AddEdge(1, 2);
AddEdge(3, 4);
AddEdge(5, 6);
AddEdge(7, 8);
AddEdge(9, 10);
int n = a.Length;
// Calling Function
Console.WriteLine(
MinimumSumConnectedComponents(a, n));
}
} |
const N = 10000; const graph = new Array(N).fill( null ).map(() => []);
const visited = new Array(N).fill( false );
// DFS function that visits all connected nodes from a given node function dfs(node, a, mini) {
// Stores the minimum
mini[0] = Math.min(mini[0], a[node]);
// Marks node as visited
visited[node] = true ;
// Traversed in all the connected nodes
for (const i of graph[node]) {
if (!visited[i]) {
dfs(i, a, mini);
}
}
} // Function to add the edges function addEdge(u, v) {
graph[u - 1].push(v - 1);
graph[v - 1].push(u - 1);
} // Function that returns the sum of all minimums of connected components of graph function minimumSumConnectedComponents(a, n) {
// Initially sum is 0
let sum = 0;
// Traverse for all nodes
for (let i = 0; i < n; i++) {
if (!visited[i]) {
const mini = [a[i]];
dfs(i, a, mini);
sum += mini[0];
}
}
// Returns the answer
return sum;
} // Driver Code for (let i = 0; i < N; i++) {
graph[i] = [];
} const a = [1, 6, 2, 7, 3, 8, 4, 9, 5, 10]; // Add edges addEdge(1, 2); addEdge(3, 4); addEdge(5, 6); addEdge(7, 8); addEdge(9, 10); const n = a.length; // Calling Function console.log(minimumSumConnectedComponents(a, n)); |
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